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Marcin Bobieski, Pawe Nurowski

Irreducible SO(3) geometry in dimension five

We consider the nonstandard inclusion of SO(3) in SO(5) associated with a 5-dimensional irreducible representation. The tensor ϒ representing this reduction is found to be given by a ternary symmetric form with special properties. A 5-dimensional manifold (M, g, ϒ) with Riemannian metric g and ternary form generated by such a tensor has a corresponding SO(3) structure, whose Gray-Hervella type classification is established using 𝔰𝔬(3)-valued connections with torsion.

Structures with antisymmetric torsions, we call them the nearly integrable SO(3) structures, are studied in detail. In particular, it is shown that the integrable models (those with vanishing torsion) are isometric to the symmetric spaces M₊ = SU(3)/SO(3), M₋ = SL(3, R)/SO(3), M₀ = ℝ⁵. We also find all nearly integrable SO(3) structures with transitive symmetry groups of dimension d > 5 and some examples for which d = 5.

Given an SO(3) structure (M, g, ϒ), we define its “twistor space” 𝕋 to be the 𝕊²-bundle of those unit 2-forms on M which span ℝ³ = 𝔰𝔬(3). The 7-dimensional twistor manifold 𝕋 is then naturally equipped with several CR and G₂ structures. The ensuing integrability conditions are discussed and interpreted in terms of the Gray-Hervella type classification.

Journal fur die reine und angewandte Mathematik (Crelles Journal), Walter de Gruyter

Print ISSN: 0075-4102
Volume: 2007, 04/2007
Pages: 51 - 93

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